Parametric Integration Revision Notes for AQA A-Level Mathematics

9.2.2 Parametric Integration

Integration of Parametric Equations

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📝Problem:

Find the area between the curve and the $x$ -axis between the given limits.

Given:

x = 1 + t, \quad y = 1 - \frac{1}{t}

Solution:

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General Formula:

The area under a curve for a parametric equation is given by:

\int y \, dx

For the given curve, the area is:

\int_{2}^{6} \left(1 - \frac{1}{t}\right) \, dx

Substituting:

Since $y$ is given in terms of t, we substitute:

y = 1 - \frac{1}{t}

Now we need to replace $dx$ with an expression in terms of $t$ .

Finding $\frac{dx}{dt}$ :

Given $x = 1 + t$ :

\frac{dx}{dt} = 1

Therefore, $dx = dt$ .

Adjusting the Integral:

Replace $dx$ with dt in the integral:

\int_{t_1}^{t_2} \left(1 - \frac{1}{t}\right) \, dt

Finding the Limits:

The limits were initially for $dx$ . We now convert them to limits for $t$ :
Lower limit: When $x = 2$ , $2 = 1 + t$ gives $t = 1$ .
Upper limit: When $x = 6$ , $6 = 1 + t$ gives $t = 5$ .
Therefore, the integral becomes:

I = \int_{1}^{5} \left(1 - \frac{1}{t}\right) \, dt

Evaluating the Integral:

Solve the integral:

I = \left[t - \ln|t|\right]_{1}^{5} = \left[(5 - \ln 5) - (1 - \cancel{\ln 1})\right]

Simplify the expression:

I = :highlight[4 - \ln 5]

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📑Example: Find the Shaded Area

Given:

x = 3t, \quad y = \sin(t), \quad t \in \mathbb{R}

Explanation:

Identifying the Limits:

In this example, no explicit limits are provided. However, it is clear that the limits lie on the $x$ -axis where $y = 0$ .
The sine function, $\sin(t)=0$ , when $t = 0$ and $t = \pi$ .

Setting Up the Integral:

To find the area, we integrate $y \, dx$ :

\int y \, dx = \int \sin(t) \, dx

We find $\frac{dx}{dt} = 3$ , which gives us $dx = 3 \, dt$ .

Adjusting the Integral:

Substitute $dx = 3 \, dt$ into the integral:

I = \int_{0}^{\pi} \sin(t) \times 3 \, dt = \int_{0}^{\pi} 3 \sin(t) \, dt

Evaluating the Integral:

Calculate the integral:

I = \left[ -3 \cos(t) \right]_{0}^{\pi}

Substitute the limits:

= (-3 \cos(\pi)) - (-3 \cos(0))

Simplify:

= 3 + 3 = :highlight[6]

Parametric Integration (AQA A-Level Mathematics): Revision Notes

9.2.2 Parametric Integration

Integration of Parametric Equations

📝Problem:

Solution:

📑Example: Find the Shaded Area

Explanation:

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